Question

How about this one...any help around?

`int dx/(sqrt(2x^2+5)`

Answer 1

`int 1/(sqrt(2x^2+5)) \ dx = 1/sqrt(2) \ sinh^(-1)(sqrt(2)/sqrt(5)x) + C`

Explanation:

We have:

`I = int 1/(sqrt(2x^2+5)) \ dx`

We can simplify and standardise the quadratic in the denominator.

Let `u=sqrt(2)/sqrt(5)x => (du)/dx = sqrt(2)/sqrt(5)`

Substituting into the integral we get:

`I = int 1/(sqrt(2(sqrt(5)/sqrt(2)u)^2+5) )\ (sqrt(5)/sqrt(2)) \ du`
`\ \ = sqrt(5)/sqrt(2) \ int 1/( sqrt(5u^2+5) ) \ du`
`\ \ = sqrt(5)/sqrt(2) \ int 1/( sqrt(5)sqrt(u^2+1) ) \ du`
`\ \ = 1/sqrt(2) \ int 1/( sqrt(u^2+1) ) \ du`

This is now a standard integral, so we have:

`I = 1/sqrt(2) \ sinh^(-1)u + C`

And, restoring the substitution, we get:

`I = 1/sqrt(2) \ sinh^(-1)(sqrt(2)/sqrt(5)x) + C`